Abelian subalgebras and ideals of maximal dimension in Lie algebras

En esta tesis se han estudiado subálgebras e ideales abelianos de álgebras de Lie, considerando dos invariantes, llamados alfa y beta, que representan el máximo entre la dimensión de todas las subálgebras abelianas (ideales para la beta) de un álgebra de Lie. Hemos desarrollado un estudio teórico en el capítulo dos, con algunos límites generales y propiedades. Después de eso, se han estudiado los casos de codimensión 1, 2 y 3. También hemos tratado la obtención de subálgebras abelianas y los ideales de varias familias específicas de álgebras de Lie resolubles. Después, hemos implementado un método algorítmico para calcular el valor de los invariantes alfa y beta, así como un representante de ellos. Y por último, mostramos algunas aplicaciones.In this thesis, we have studied abelian subalgebras and ideals of Lie algebras by considering two invariants, named alpha and beta, which represent the maximum among the dimension of all the abelian subalgebras (ideals for beta) of a Lie algebra. We have developed a theoretical study in Chapter two with some general bounds and properties. After that, we have studied the cases of codimension 1, 2 and 3. We have also dealt with the obtainment of abelian subalgebras and ideals in several specific families of solvable lie algebras. Then, we have implemented an algorithmic method to compute the value of alpha and beta invariants, as well as a representative for them. Finally, some applications are shown.Premio Extraordinario de Doctorado U

Graph operations and Lie algebras

This paper deals with several operations on graphs and combinatorial structures linking them with their associated Lie algebras. More concretely, our main goal is to obtain some criteria to determine when there exists a Lie algebra associated with a combinatorial structure arising from those operations. Additionally, we show an algorithmic method for one of those operations.Ministerio de Ciencia e InnovaciónFondo Europeo de Desarrollo Regiona

Pedro de Lucuce y Ponce y las instituciones matemático-militares españolas del siglo XVIII

Dos años después, vuelve a la sección de Historia de La Gaceta de la RSME el tema matemático-militar durante el siglo XVIII español, tema y periodo dominados por la figura del marino Jorge Juan y Santacilia (1713-1773), nacido hace justo doscientos años. El artículo que sigue nos acerca a otra figura relevante, el militar Pedro de Lucuce y Ponce (1692-1779), cuya vida exponen con acierto los autores. Se refieren también a su obra, escrita o inédita, en la que la matemática tiene un papel importante, como corresponde a un ingeniero militar con una larga experiencia en la formación científica y técnica de los oficiales. Se invita al lector a contrastar esta dedicación a la matemática en el seno de la vida activa, productiva y ordenada del militar con la no menos activa, pero radicalmente desordenada, de su coetáneo Diego Torres de Villarroel (1694-1770), el singular catedrático de matemáticas de la Universidad de Salamanca, cuya autobiografía, fácil de localizar, es una lectura recomendable

Relations between combinatorial structures and Lie algebras: centers and derived Lie algebras

In this paper, we study how two important ideals of a given Lie algebra g (namely, the center Z(g) and the derived Lie algebra D(g)) can be translated into the language of Graph Theory. In this way, we obtain some criteria and characterizations of these ideals using Graph Theory.Ministerio de Ciencia e InnovaciónFondo Europeo de Desarrollo Regiona

Computing abelian subalgebras for linear algebras of upper-triangular matrices from an algorithmic perspective

In this paper, the maximal abelian dimension is algorithmically and computationally studied for the Lie algebra hn, of n×n upper-triangular matrices. More concretely, we define an algorithm to compute abelian subalgebras of hn besides programming its implementation with the symbolic computation package MAPLE. The algorithm returns a maximal abelian subalgebra of hn and, hence, its maximal abelian dimension. The order n of the matrices hn is the unique input needed to obtain these subalgebras. Finally, a computational study of the algorithm is presented and we explain and comment some suggestions and comments related to how it works

Minimal faithful upper-triangular matrix representations for solvable Lie algebras

A well-known result on Lie Theory states that every finite-dimensional complex solvable Lie algebra can be represented as a matrix Lie algebra, with upper-triangular square matrices as elements. However, this result does not specify which is the minimal order of the matrices involved in such representations. Hence, the main goal of this paper is to revisit and implement a method to compute both that minimal order and a matrix representative for a given solvable Lie algebra. As application of this procedure, we compute representatives for each solvable Lie algebra with dimension less than 6.Ministerio de Economía, Industria y CompetitividadFondo Europeo de Desarrollo Regiona

An algorithm to compute abelian subalgebras in linear algebras of upper-triangular matrices

This paper deals with the maximal abelian dimension of the Lie algebra hn, of nxn upper-triangular matrices. Regarding this, we obtain an algorithm which computes abelian subalgebras of hn as well as its implementation (and a computational study) by using the symbolic computation package MAPLE, where the order n of the matrices in hn is the unique input needed. Let us note that the algorithm also allows us to obtain a maximal abelian subalgebra of hn

The computation of Abelian subalgebras in low-dimensional solvable Lie algebras

The main goal of this paper is to compute the maximal abelian dimension of each solvable nondecomposable Lie algebra of dimension less than 7. To do it, we apply an algorithmic method which goes ruling out non-valid maximal abelian dimensions until obtaining its exact value. Based on Mubarakzyanov and Turkowsky’s classical classifications of solvable Lie algebras (see [13] G.M. Mubarakzyanov: Classification of real structures of Lie algebras of fifth order. Izv. Vyss. Ucebn. Zaved. Matematika 3:34, 1963, pp. 99-106. and [19] P. Turkowski: Solvable Lie algebras of dimension six. J. Math. Phys. 31, 1990, pp. 1344-1350) and the classification of 6-dimensional nilpotent Lie algebras by Goze and Khakimdjanov [7] M. Goze and Y. Khakimdjanov: Nilpotent and solvable Lie algebras. In M. Hazewinkel (ed.): Handbook of Algebra Vol 2. Elsevier, Amsterdam, 2000, pp. 615–664, we have explicitly computed the maximal abelian dimension for the algebras given in those classifications

Obtaining combinatorial structures associated with low-dimensional Leibniz algebras

In this paper, we analyze the relation between Leibniz algebras and combinatorial structures. More concretely, we study the properties to be satisfied by (pseudo)digraphs so that they are associated with low-dimensional Leibniz algebras. We present some results related to this association and show an algorithmic method to obtain them, which has been implemented with Maple